Difference between revisions of "2014 USAJMO Problems/Problem 2"
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(b) Line <math>OH</math> intersects segments <math>AB</math> and <math>AC</math> at <math>P</math> and <math>Q</math>, respectively. Denote by <math>s</math> and <math>t</math> the respective areas of triangle <math>APQ</math> and quadrilateral <math>BPQC</math>. Determine the range of possible values for <math>s/t</math>. | (b) Line <math>OH</math> intersects segments <math>AB</math> and <math>AC</math> at <math>P</math> and <math>Q</math>, respectively. Denote by <math>s</math> and <math>t</math> the respective areas of triangle <math>APQ</math> and quadrilateral <math>BPQC</math>. Determine the range of possible values for <math>s/t</math>. | ||
==Solution== | ==Solution== | ||
| + | We draw a diagram to not lose points: | ||
| + | |||
| + | '''Part a''' | ||
| + | |||
| + | '''Part b''' | ||
Revision as of 19:45, 29 April 2014
Problem
Let 
be a non-equilateral, acute triangle with $\angle A=60\textdegrees$ (Error compiling LaTeX. Unknown error_msg), and let 
and 
denote the circumcenter and orthocenter of , respectively.
(a) Prove that line 
intersects both segments 
and 
.
(b) Line intersects segments and at 
and 
, respectively. Denote by 
and 
the respective areas of triangle 
and quadrilateral 
. Determine the range of possible values for 
.
Solution
We draw a diagram to not lose points:
Part a
Part b
